GSL of Thermodynamics for Holographic Dark Energy and … · 2016. 3. 2. · Title: GSL of...

5o Encuentro COSMOCONCE: Cosmología y Gravitación enConcepción

GSL of Thermodynamics for HolographicDark Energy and Cosmological Interaction

F. Arévalo , P. Cifuentes, F. Peñ[email protected]

Departamento de Ciencias Físicas, Facultad de IyC, Universidad deLa Frontera.

April 7, 2016, Concepción, Chile.

1

Content

Introduction

Generalized Second LawHorizons and Dark EnergyKnown horizon and the GSLRicci and Ricci-like scenarios

F. Arevalo | GSL for HDE and Q

2

Cosmology in the context of General Relativity

Einstein Field Eqs.

Rµν −12

R gµν = κTµν

gµν is the metric, Rµν is the Ricci tensor,R is the Ricci scalar Tµν is the energymomentum tensor.

Cosmological PrincipleThe Universe is homogeneous and isotropic. The most general metric thatfulfills this is the Friedmann-Robertson-Walker(FRW) metric

ds2 = dt2 − a(t)2(

dr 2

1 − kr 2 + r2(

dθ2 + sin2 θdφ2))

a(t) is the scale factor, (r , θ, φ) arethe spatial coordinates, t is thecosmic time, p the pressure and ρthe total energy density.

Idealized Matter Content

Tµν = (ρ+ p)uµuν − p gµν


3


4

The SCM is a Cold Dark Matter (CDM) component with no pressure(pDM = 0) and a Cosmological constant component (pΛ = −ρΛ)

Equation of Motion ΛCDMρ̇DM + 3HρDM = 0→ ρDM = ρ0DMa−3

ρ̇Λ = 0→ ρΛ = ρ0Λ

Defining Ω0i =ρ0i

3H20

r0 =Ω0DMΩ0Λ≈ O(1)

Modifications of Dark EnergyI Cosmological InteractionI Holographic Dark EnergyI Many others...

Interacting Dark Sector

ρ̇DE + 3H (ρDE + pDE ) = −Q,ρ̇DM + 3H (ρDM + pDM ) = Q.

Ricci-like ρDE = 3αH2 + 3βḢ


5

Outline

Introduction



6

Generalized Second Law of Thermodynamics

To examine the thermodynamic behavior of a cosmological scenario,the universe (in our case the interacting dark sector) can beconsidered as a thermodynamical system with a certain boundary Lin a brief state of equilibrium.

TDE dSDE = pDE dV + dEDE , EDE ≡ 4π3 L3ρDE ,

TDM dSDM = pDM dV + dEDM , EDM ≡ 4π3 L3ρDM .

Its associated temperature is given as Th ≡ 12πL , according to [8] andreferences therein. The entropy associated with the horizon isSh ≡ 8π2L2.

The total entropy rate is Ṡtot = ṠDE + ṠDM + Ṡh

Ṡtot16π2L

=(

1− ḢL2)(

L̇− HL)

+ HL.


7

Outline

Introduction



8

Horizon related to the length scale ρDE ≡ 3c2

L2

We write the GSL using q ≡ −(

1 + ḢH2)

and r ≡ ρDMρDE =H2L2

c2 − 1

HṠtot16π2c2 =

[1 + c2(1 + r)(q + 1)

] [ r ′2 + (1 + r)q

]+ (1 + r).

We can also write it in terms of PDE =pDEH2 and Γ =

Q3H3

S′tot4π2L2 = 4 +

[(3 + PDE ) c

2(1 + r) + 2]× [1 + (1 + r) (PDE + Γ)] ,

Using auxiliary functions

f (a) ≡ −PDE − 1(1+r) , ,

g(a) ≡ −4(1+r)((3+PDE )c2(1+r)+2)

,

S′tot4π2L2 = −

4g(a) (Γ− f (a)− g(a))

If g(a) > 0, then we obtainΓ < f (a) < f (a) + g(a). On theother hand if g(a) < 0, then weobtain f (a) > Γ > f (a) + g(a).


9

Outline

Introduction



10

Known horizon and the GSL

There are known horizon in the literature for holographic dark energyscenarios, we shall analyze briefly the GSL on three of them:The Hubble horizon LH in the FRW flat universe is given by

LH ≡ H−1 → H ṠH16π2 =(

H−2Ḣ)2

The cosmological event horizon LE is defined as

LE ≡ a∫∞

tdta →

ṠE16π2LE

= 2c2(1 + r)2r ′ − c2(1 + r)− 1

Now, we consider a linear combination of LH and LE assumed by [9],while investigating thermodynamics and the phantom barrier,

LHE ≡ λHLH + λELE →ṠHE

16π2LHE=

[1− c2 r

′

2+ c2(r + 1)

](λH − λE )

+[(λH − λE )2 + 1

]c√

r + 1 ,


11

Outline

Introduction



12

Ricci Dark Energy scenario

We consider that since the dark energy is related to the horizon in anholographic context, and Ricci and Ricci-like scenarios consider thedark energy as an specific Ansatz, we shall study the possibility ofusing such Ansatz as an expression for a surface representing thehorizon in a thermodynamical context.

The Ricci dark energy

ρDE ≡ 3c2(2H2 + Ḣ) ←→ pDE = H2 − 23c2 ρDE

The auxiliary parameters are f(a)=−1 + 2−c2

c2(1+r) , g(a)=-1

c2(1+r)2 , thenI if c2 > 22+r , then interaction could be negativeI if c2 < (1+2r)(2+r)(1+r) , the interaction Γ should always be positive at

late times.The total entropy rate is given as

HṠR8π2c2(1+r) = 2c

2(1 + r) [1 + (r + 1) (1 + Γ)]− 2(1 + 2r)


13

Ricci Dark Energy

Now, we consider the entropy change without the interaction explicitly,but rather in terms of r and r ′, given as

HṠR16π2 =

[1− 1c2 + 2(1 + r)

] [ r ′2 −

1c2 + (1 + r)

]+ (1 + r).

We can write the pressure as PDE = −[

2c2(1+r) − 1

], that indicates

that the evolution of r could be determined by the evolution of PDE ,relation obtained from the holographic context. Then the interactioncan be written as

Γ =r ′ + r

[ 2c2 − (1 + r)

](1 + r)2

, (1)

a function of (r , r ′). In our case the sign of Γ remains undeterminedso far. Using r as a function of PDE and its derivative in the interaction,we obtain

Γ =c2

2

[P ′

DE− PDE

(2c2− 1 + PDE

)], (2)


14

Observational Constraints for Ricci DE

Figure : Phase-space of (r , c2) considering zones where the interaction Qrespects the GSL for Ricci Dark Energy. The shaded area is where thepressure is negative at late times. The dots are pairs of (r0, c2) obtained withobservational data from [?] and references therein, and the vertical lines forr0 are indicated for comparison.


15

Ricci-like Dark Energy scenario

Ricci-like dark energy

ρDE ≡ 3(αH2 + βḢ) ←→ pDE = − 23β ρDE + (2α− 3β)H2/β

Considering this equation in terms of r , we obtain

HṠRl8π2(1+r) = 2 +

[1− 1β (1− α(1 + r))

] [1− 2β + (1 + r)

(2α−3ββ + Γ

)]the auxiliary parameters aref (a) = 2−ββ(1+r) −

2α−3ββ , g(a) =

−2β/(1+r)β−1+α(1+r)

The sign of interaction in Ricci-like HDE can be negative or positivewhile respecting GSL, according to the ranges obtained fromreplacing (f ,g) inequalities in terms of r , α and β.


16

Ricci-like Dark Energy

β2H ṠRl16π2 = β2(r + 1) + [β − (1− α(r + 1))]

[β2 r′ − 1 + (α− β)(1 + r)

]In this context by choosing a horizon we are inherently choosing apressure, as a function of r and the model parameters

PDE = − [2/(1 + r)− (2α− 3β)] /β.

The result obtained indicates that the evolution of r could bedetermined by the evolution of PDE , relation obtained from theholographic context. Then, the interaction can be written as

Γ =β r ′ + r (2− (2α− 3β)(1 + r))

β(1 + r)2,

Using r(PDE ) and its derivative in the interaction

Γ =(βP ′

DE− PDE (2(1− α) + β(3 + PDE ))

)/2.

It is equivalent to study the expression for interaction described by ras it is the interaction as a function of the pressure PDE .


17

Observational Constraints for Ricci-like

We studied several cases for the entropy rate function

0.5 1.0 1.5 2.0

50

100

150

200

0.5 1.0 1.5 2.0

-3-2-1

1

2

wΓq


18

Figure : The graph on the left represents the relation between the GSL andthe interaction for a specific r0 in the phase space of (α, β) and the graph onthe right is the entropy change and the top is the deceleration parameter andinteraction for a specific holographic model, case III of [2].


19

Final Remarks

We concluded thatI We propose the HDE Ansatz as an horizon -like radius to study

features of the cosmological interaction using thermodynamics.I We consider the interaction with an unknown sign, generalizing

previous studies where the interaction is generally given andpositive

I The variation with respect to the cosmic time of the total entropycan be positive or negative depending on the horizon chosenwhen this horizon includes a term of Ḣ that inherentlydetermines the pressure.

I For these cases the cosmic coincidence problem is alleviated,given that when the pressure is a variable function, r is also avariable function.

I The Ricci-like HDE is allowed to be negative and respect GSLbut only for a certain range, noticeable in Fig 2

I More work in sign-changeable interactions could be exploredF. Arevalo | GSL for HDE and Q

20

References

[1] L. P. Chimento, Phys. Rev. D 81, 043525 (2010).

[2] F. Arevalo, P. Cifuentes, S. Lepe and F. Pena, Astrophys. Space Sci.352, 899 (2014)

[3] S. Lepe, F. Pena and F. Torres, Phys. Rev. D 91, no. 2, 024023 (2015)

[4] H.Wei, Nucl.Phys. B845 (2011) 381-392.

[5] Y. -H. Li and X. Zhang, “Running coupling: Does the coupling betweendark energy and dark matter change sign ..."’

[6] M. Khurshudyan, arXiv:1302.1220 [gr-qc].

[7] F. Arevalo, A. P. R. Bacalhau and W. Zimdahl, Class. Quant. Grav. 29,235001 (2012)

[8] M. Jamil, E. N. Saridakis and M. R. Setare, Phys. Rev. D 81, 023007(2010)

[9] H. M. Sadjadi and M. Honardoost, Phys. Lett. B 647, 231 (2007)


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IntroductionGeneralized Second LawHorizons and Dark EnergyKnown horizon and the GSLRicci and Ricci-like scenarios

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